Recent zbMATH articles in MSC 57N https://www.zbmath.org/atom/cc/57N 2021-04-16T16:22:00+00:00 Werkzeug Topology change of level sets in Morse theory. https://www.zbmath.org/1456.37061 2021-04-16T16:22:00+00:00 "Knauf, Andreas" https://www.zbmath.org/authors/?q=ai:knauf.andreas "Martynchuk, Nikolay" https://www.zbmath.org/authors/?q=ai:martynchuk.nikolay This paper considers Morse functions $$f \in C^2(M, \mathbb{R})$$ on a smooth $$m$$-dimensional manifold without boundary. For such functions and for every critical point $$x \in M$$ of $$f$$, the Hessian of $$f$$ at $$x$$ is nondegenerate. The authors are interested in the topology of the level sets $$f^{-1}(a) = \partial{M^a}$$ where $$M^a = \{ x \in M : f(x) \le a\}$$ is a so-called sublevel set. The authors address the question of under what conditions the topology of $$\partial{M^a}$$ can change when the function $$f$$ passes a critical level with one or more critical points. Change in topology'' here means roughly that if $$a$$ and $$b$$ are regular values with $$a < b$$, then $$H_k(\partial{M^a};G) \ne H_k(\partial{M^b};G)$$ where $$k$$ is an integer and the $$H_k$$ are homology groups over some abelian group $$G$$ when the function $$f$$ passes through a level with one or more critical points. A motivation for this question derives from the $$n$$-body problem: does the topology of the integral manifolds always change when passing through a bifurcation level? The first part of the paper considers level sets of abstract Morse functions that satisfy the Palais-Smale condition and whose level sets have finitely generated homology groups. They show that for such functions the topology of $$f^{-1}(a)$$ changes when passing a single critical point if the index of the critical point is different from $$m/2$$ where $$m$$ is the dimension of the manifold $$M$$. Then they move on to consider the case where $$M$$ is a vector bundle of rank $$n$$ over a manifold $$N$$ of dimension $$n$$, and where (up to a translation) $$f$$ is a Morse function that is the sum of a positive definite quadratic form on the fibers and a potential function that is constant on the fibers. Here the authors have in mind Hamiltonian functions $$H$$ on the cotangent bundle of the base manifold $$N$$. In this case they show that the topology of $$H^{-1}(h)$$ always changes when passing a single critical point if the Euler characteristic of $$N$$ is not $$\pm 1$$. Finally the authors apply their results to examples from Hamiltonian and celestial mechanics, with emphasis on the planar three-body problem. There they show that the topology always changes for the planar three-body problem provided that the reduced Hamiltonian is a Morse function with at most two critical points on each level set. Reviewer: William J. Satzer Jr. (St. Paul) Models of simply-connected trivalent 2-dimensional stratifolds. https://www.zbmath.org/1456.57014 2021-04-16T16:22:00+00:00 "Gómez-Larrañaga, J. C." https://www.zbmath.org/authors/?q=ai:gomez-larranaga.jose-carlos "González-Acuña, F." https://www.zbmath.org/authors/?q=ai:gonzalez-acuna.francisco-javier "Heil, Wolfgang" https://www.zbmath.org/authors/?q=ai:heil.wolfgang-h Summary: Trivalent 2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where three sheets meet. We develop operations on their associated labeled graphs that will effectively construct from a single vertex all graphs that represent simply connected trivalent 2-stratifolds. Invariants and TQFT's for cut cellular surfaces from finite groups. https://www.zbmath.org/1456.57026 2021-04-16T16:22:00+00:00 "Bragança, Diogo" https://www.zbmath.org/authors/?q=ai:braganca.diogo "Picken, Roger" https://www.zbmath.org/authors/?q=ai:picken.roger-f Summary: We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the 1-cells with elements of a finite group, subject to a flatness'' condition for each 2-cell. These invariants are also described in a TQFT setting, which is not the same as the usual 2-dimensional TQFT framework. We study the properties of functions which arise in this context, associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology, including properties which come from invariance under the Hatcher-Thurston moves on pants decompositions. Results on the homotopy type of the spaces of locally convex curves on $\mathbb{S}^3$. https://www.zbmath.org/1456.57024 2021-04-16T16:22:00+00:00 "Alves, Emília" https://www.zbmath.org/authors/?q=ai:alves.emilia "Saldanha, Nicolau C." https://www.zbmath.org/authors/?q=ai:saldanha.nicolau-corcao A smooth curve $$\gamma:[0,1]\to \mathbb{S}^3$$ in $$4$$-dimensional Euclidean space $$\mathbb{R}^4$$ with image on the sphere $$\mathbb{S}^3$$, is said to be locally convex, if the set of vectors $${\gamma}(t),{\gamma}'(t), {\gamma}''(t),{\gamma}'''(t)$$ is a positive basis in $$\mathbb{R}^4$$ for all $$t\in[0,1]$$. By the Gram-Schmidt procedure, we can turn this basis into an orthonormal basis and thereby associate a Frenet frame curve $$\mathcal{F}_{\gamma}: [0,1]\to SO_4$$ in the special orthogonal group $$SO_4$$ to a locally convex curve. For any matrix $$Q\in SO_4$$, let $$\mathcal{L}\mathbb{S}^3(Q)$$ denote the space of all locally convex curves $$\gamma:[0,1]\to \mathbb{S}^3$$ where $$\mathcal{F}_{\gamma}(0)=I$$ (the identity matrix) and $$\mathcal{F}_{\gamma}(1)=Q$$. It was proved by [\textit{N. C. Saldanha} and \textit{B. Shapiro}, J. Singul. 4, 1--22 (2012; Zbl 1292.58002)] that there are at most 3 different homeomorphism types among the path components of $$\mathcal{L}\mathbb{S}^3(Q)$$. But otherwise very little seems to be known about these components and their generalizations to higher dimensional spheres $$\mathbb{S}^n$$. In the present paper the authors prove several interesting theorems on the homotopy and homology of these path components, in particular for the case $$Q=-I$$. The results are technical and cannot be given in detail. Reviewer: Vagn Lundsgaard Hansen (Lyngby)